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Non-positively curved complexes

Final Activity Report Summary - NPC COMPLEXES (Non-positively curved complexes)

The project concerns geometric group theory. We were studying group acting geometrically on cell complexes with various combinatorial non-positive curvatures-e.g. simplicial non-positive curvature (i.e. systolic groups) and CAT(0) cube complexes. In particular we were studying boundaries of such groups. The main outcomes of this fellowship are the two papers:
[1] (with Piotr Przytycki) 'Boundaries of systolic groups', submitted.
[2] 'On simplicial non-positive curvature', in preparation.

In [1] we construct the boundary (in the sense of the Bestvina Z-structure) of systolic groups. Existence of such a boundary is conjectured for a large class of groups but up to now it is known only for few classes of groups. As a corollary we get e.g. the Novikov conjecture for torsion-free systolic groups.

In [2] we introduce a new notion of a combinatorial non-positive curvature. This theory includes in particular systolic and CAT(0) cubical worlds. As an application we can extend some results obtained earlier for systolic groups.

As for the training aspect of the project the most important was initiating of the regular studies in the domain of analysis on the boundaries of hyperbolic groups.